On the two-dimensional Jacobian conjecture: Magnus' formula revisited, I

TL;DR

This paper revisits Magnus' formula in the context of the two-dimensional Jacobian conjecture, extending the formula and applying it to prove a special case of the conjecture.

Contribution

It generalizes Magnus' formula and uses it to establish a partial result towards the two-dimensional Jacobian conjecture.

Findings

Extended Magnus' formula to a more general setting.

Proved a special case of the two-dimensional Jacobian conjecture.

Clarified relations between homogeneous degree components in polynomial maps.

Abstract

Let $K$ be an algebraically closed field of characteristic 0. When the Jacobian $({\partial f}/{\partial x})({\partial g}/{\partial y}) - ({\partial g}/{\partial x})({\partial f}/{\partial y})$ is a constant for $f,g\in K[x,y]$, Magnus' formula from [A. Magnus, Volume preserving transformations in several complex variables, Proc. Amer. Math. Soc. 5 (1954), 256--266] describes the relations between the homogeneous degree pieces $f_i$'s and $g_i$'s. We show a more general version of Magnus' formula and prove a special case of the two-dimensional Jacobian conjecture as its application.